In the number series 10, 21, 54, 109 and 189, one term is wrong. Which number is incorrect if the series is meant to follow a regular pattern in the differences?

Introduction / Context:
This question asks you to detect the wrong term in a number series. All the terms are fairly close together and the series appears to increase in a non linear way. Often, such questions hide a simple pattern in the first differences or second differences between successive terms. The goal is to find a smooth rule that works for all but one term.

Given Data / Assumptions:
The series is:

• 10
• 21
• 54
• 109
• 189

We assume that the intended series is generated by adding regularly increasing differences, and exactly one of the given terms does not fit this intended pattern.

Concept / Approach:
A good starting point is to calculate the difference between consecutive terms. If these differences themselves form an arithmetic progression or follow some simple rule, then any term that breaks this rule can be identified as incorrect. Here we will check whether the differences form a simple arithmetic sequence.

Step-by-Step Solution:
Step 1: Compute the first differences.From 10 to 21 the difference is 21 − 10 = 11.From 21 to 54 the difference is 54 − 21 = 33.From 54 to 109 the difference is 109 − 54 = 55.From 109 to 189 the difference is 189 − 109 = 80.Step 2: Observe the first three differences: 11, 33 and 55. These differ by 22 each time: 33 − 11 = 22 and 55 − 33 = 22.Step 3: If the pattern continued, the next difference should be 55 + 22 = 77.Step 4: Therefore, the correct last term should be 109 + 77 = 186, not 189.

Verification / Alternative check:
The intended difference pattern is therefore 11, 33, 55, 77, a sequence where each difference increases by 22. Rebuilding the series with this pattern gives 10, 21, 54, 109 and 186. This reconstructed list is completely consistent. Since 189 does not fit this clean difference pattern, it must be the wrong entry in the original series.

Why Other Options Are Wrong:
If 21, 54 or 109 were wrong, it would be impossible to create a smooth arithmetic progression of differences starting from 10 and still keep the remaining terms lined up neatly. Only by changing 189 to 186 do we obtain a consistent and simple rule for all differences. Therefore these other numbers cannot reasonably be the incorrect term.

Common Pitfalls:
Some students try to find a direct multiplication pattern and may get stuck. For many series problems, checking first and second differences is the most reliable method. Once you notice that the first three differences form a regular pattern, it becomes much easier to spot which subsequent difference breaks the rule and leads to the wrong term.

Final Answer:
The number that does not fit the intended difference pattern and is therefore wrong is 189.